Algebraic K-theory and trace invariants

The cyclotomic trace of B"okstedt-Hsiang-Madsen, the subject of B"okstedt’s lecture at the congress in Kyoto, is a map of pro-abelian groups K_(A) -> TR_^.(A;p) from Quillen’s algebraic K-theory to a topological refinement of Connes’ cyclic homology. Over the last decade, our understanding of the target and its relation to K-theory has been significantly advanced. This and possible future development is the topic of my lecture.